Über die Häufigkeit großer Differenzen konsekutiver Primzahlen

Abstract

The following theorem is going to be proved. Letp _m be them-th prime and putd _m :=p _m+1−p _m . LetN(σ,T), 1/2≤σ≤1,T≥3. denote the number of zeros ϱ=β+iγ of the Riemann zeta function which fulfill β≥σ and |γ|≤T. Letc≥2 andh≥0 be constants such thatN(σ,T)≪T ^c(1−σ) (logT)^h holds true uniformly in 1/2≤σ≤1. Let ε>0 be given. Then there is some constantK>0 such that